Closed Legendre geodesics in Sasaki manifolds

Knut Smoczyk

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Submission date: 05. Sep. 2002
published in: New York journal of mathematics, 9 (2003), p. 23-47 (electronic) 
Bibtex
MSC-Numbers: 53C44, 53C42
Keywords and phrases: legendre, curve shortening, geodesic, sasaki

Abstract:
If is a Legendre submanifold in a Sasaki manifold, then the mean curvature flow does not preserve the Legendre condition. We define a kind of mean curvature flow for Legendre submanifolds which slightly differs from the standard one and then we prove that closed Legendre curves L in a Sasaki space form M converge to closed Legendre geodesics, if and , where denotes the sectional curvature of the contact plane and k, are the curvature respectively the rotation number of L. If , we obtain convergence of a subsequence to Legendre curves with constant curvature. In case and if the Legendre angle of the initial curve satisfies , then we also prove convergence to a closed Legendre geodesic.